Shakhar Smorodinsky

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Locally Delaunay Geometric Graphs
Shakhar Smorodinsky ETH Zurich Joint work
with Rom Pinchasi, MIT
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Geometric Graphs
A Geometric Graph is G(V,E) embedding in R2
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K-locally Delaunay graphs
A Geometric graph is k-locally Delaunay if Can
be embedded s.t. every edge is isolated from its
(k)-neighbors by some disc
1-locally Delaunay
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Problem 1
What is the maximum edge complexity of
a 1-locally delaunay graph G(V,E)?
Motivation?
Topology Control for Sensor Networks
First observation S. Kapoor, X.Y. Li 03 G
cannot contain a K3,3. Hence E O(n5/3)
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G contains no K3,3. Hence E O(n5/3) Kovari,
Sos, Turan Proof
As a matter of fact if G can be embeded without
a self-crossing C4 then E O(n8/5) .
Pinchasi, Radoicic 03.
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Next improvementCan G contain a K2,2 ?
Yes!
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Our contribution
Thm Pinchasi, S If G(V,E) is 1-locally
Delaunay then EO(n3/2)

• Lets assume many things
• All edges makes a small angle with the vector
  (0,1).
• 2) All edges cross the x-axis
• 3) For every edge e, the witnessing disc is such
  that its center is left to e

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If G(V,E) is 1-locally Delaunay then
EO(n3/2) Proof
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If G(V,E) is 1-locally Delaunay then
EO(n3/2) Proof (cont)
Under these assumptions
G contains no K2,2
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If G(V,E) is 1-locally Delaunay then
EO(n3/2) Proof (cont)
Assume G contains K2,2
A contradiction
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Thm Pinchasi, S If G(V,E) is 2-locally
Delaunay then EO(n) Remark
First observation G contains no self crossing
copy of P3
Hence by Pach, Pinchasi, Tardos, Toth 03 E
O(n log n)
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If G(V,E) is 2-locally Delaunay then
EO(n) Proof

• Lets assume small angles between edges
• Remove the (upper, lower) right most and left
  most edge from every vertex

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If G(V,E) is 2-locally Delaunay then
EO(n) Proof (cont)
Claim No edge survived!!!
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Claim No edge survived!!!